2023.6

11 A rock falls from rest through a small distance $s$ to the surface of Mars. The rock hits Mars with velocity $v_{\mathrm{M}}$.
Another rock falls from rest through distance $s$ to the surface of Earth and hits Earth with velocity $v_{\mathrm{E}}$.

Calculate the ratio $\frac{v_{\mathrm{M}}}{v_{\mathrm{E}}}$.
acceleration due to gravity on Mars $=0.38 \mathrm{~g}$

Question

2023.6

11 A rock falls from rest through a small distance $s$ to the surface of Mars. The rock hits Mars with velocity $v_{\mathrm{M}}$.
Another rock falls from rest through distance $s$ to the surface of Earth and hits Earth with velocity $v_{\mathrm{E}}$.

Calculate the ratio $\frac{v_{\mathrm{M}}}{v_{\mathrm{E}}}$.
acceleration due to gravity on Mars $=0.38 \mathrm{~g}$

Accepted Answer

∻Solution∻ ‖ a ⋮ The velocity of an object in free fall can be calculated using the equation $$v = gt$$, where $$g$$ is the acceleration due to gravity. For Mars, $$g_M = 0.38g_E$$, where $$g_E = 9.8 \, \mathrm{m/s^2}$$. Thus, $$g_M = 0.38 imes 9.8 \, \mathrm{m/s^2} = 3.724 \, \mathrm{m/s^2}$$. The time taken to fall distance $$s$$ on Mars is given by $$s = \frac{1}{2} g_M t_M^2$$, leading to $$t_M = \sqrt{\frac{2s}{g_M}}$$. Similarly, for Earth, $$t_E = \sqrt{\frac{2s}{g_E}}$$. ‖ ‖ b ⋮ The final velocities can be expressed as $$v_M = g_M t_M$$ and $$v_E = g_E t_E$$. Substituting the expressions for time, we have: $$v_M = g_M \sqrt{\frac{2s}{g_M}} = \sqrt{2sg_M}$$ and $$v_E = g_E \sqrt{\frac{2s}{g_E}} = \sqrt{2sg_E}$$. Now, we can find the ratio: $$\frac{v_M}{v_E} = \frac{\sqrt{2sg_M}}{\sqrt{2sg_E}} = \sqrt{\frac{g_M}{g_E}}$$. ‖ ‖ c ⋮ Substituting the values of $$g_M$$ and $$g_E$$, we get: $$\frac{v_M}{v_E} = \sqrt{\frac{0.38g_E}{g_E}} = \sqrt{0.38} \approx 0.616$$. ‖ ∻Answer∻ ⚹ The ratio of the velocities is approximately 0.616 ⚹ ∻Key Concept∻ ⚹ Free Fall and Velocity Calculation: The velocity of an object in free fall can be calculated using the acceleration due to gravity and the time of fall. The ratio of velocities can be derived from the gravitational accelerations of different celestial bodies. ⚹ ∻Explanation∻ ⚹ The ratio of the final velocities of the rocks falling on Mars and Earth is derived from their respective gravitational accelerations, showing how gravity affects falling objects differently on different planets. ⚹

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